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%\author{王立庆（2019级数学与应用数学1班）}
\author{学号 \underline{\hspace{4cm}} 姓名  \underline{\hspace{4cm}} }
%\title{高等代数第六章：向量空间}
\title{第九章二次型考试 }
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\date{2023年5月31日}

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\begin{document}

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\begin{enumerate}

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\item %第1题
设实二次型 $q(x_1, x_2, x_3)=x_1^2+2x_2^2+x_3^2+2x_1x_2+6x_1x_3+8x_2x_3$. 
\begin{enumerate}
\item  使用配方法，将 $q$ 化为实二次型的标准型，写出使用的变量代换。
\item  求实二次型 $q$ 的秩与符号差，判断 $q$ 是否为正定二次型。
\item  求实对称阵 $A$ 使得 $q(x_1,x_2,x_3)=X^tAX$. 
\item  计算矩阵 $A$ 的顺序主子式，判断 $A$ 是否为正定矩阵。
\end{enumerate}

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\item  %第2题
设实二次型 $q(x_1,x_2)=-2x_1^2+8x_1x_2-2x_2^2$. 
\begin{enumerate}
\item  求实对称阵 $A$ 使得 $q(x_1,x_2)=X^tAX$. 
\item  求正交矩阵 $U$ 使得 $U^tAU$ 为对角矩阵。
\item  画出二次曲线 $q(x_1,x_2)=4$ 的图像，写出主轴的方程。
\end{enumerate}

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\end{enumerate}


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